\magnification=1200
\hsize=4in
\overfullrule=0pt
\input amssym
%\def\frac#1 #2 {{#1\over #2}}
\def\emph#1{{\it #1}}
\def\em{\it}
\nopagenumbers
\noindent
%
%
{\bf V. Anuradha, Chinmay Jain, Jack Snoeyink and Tibor Szab{\'o}}
%
%
\medskip
\noindent
%
%
{\bf How Long Can a Graph be Kept Planar?}
%
%
\vskip 5mm
\noindent
%
%
%
%
The graph (non-)planarity game is played on the complete graph~$K_n$
between an Enforcer and an Avoider, each of whom take one edge per
round. The game ends when the edges chosen by Avoider form a
non-planar subgraph. We show that Avoider can play for $3n-26$
turns, improving the previous bound of $3n-28\sqrt n$.
\bye